Regularized inner products of meromorphic modular forms and higher Green's functions

In this paper we study generalizations of quadratic form Poincar\'e series, which naturally occur as outputs of theta lifts. Integrating against them yields evaluations of higher Green's functions. For this we require a new regularized inner product, which is of independent interest

Bounds for canonical Green's functions at cusps

Let $\Gamma$ be a cofinite Fuchsian subgroup. The canonical Green's function associated with $\Gamma$ arises in Arakelov theory when establishing asymptotics for Arakelov invariants of the modular curve associated with some congruence subgroup of level $N$ with a positive integer $N$. More precisely, in the known cases, canonical Green's functions at certain cusps contribute to the analytic part of the asymptotics for the self-intersection of the relative dualizing sheaf. In this article, we prove canonical Green's function of a cofinite Fuchsian subgroup at cusps bounded by the scattering constants, the Kronecker limit functions, and the Selberg zeta function of the group $\Gamma$. Then as an application, we prove an asymptotic expression of the canonical Green's function associated with $\Gamma_0(N)$, for any positive integer $N$

Special values of zeta functions and areas of triangles

In this snapshot we give a glimpse of the interplay of special values of zeta functions and volumes of triangles. Special values of zeta functions and their generalizations arise in the computation of volumes of moduli spaces (for example of Abelian varieties) and their universal spaces

RIEMANN-ROCH ISOMETRIES IN THE NON-COMPACT ORBIFOLD SETTING

International audienceWe generalize work of Deligne and Gillet-Soulé on a functorial Riemann-Roch type isometry, to the case of the trivial sheaf on cusp compactifications of Riemann surfaces Γ\H, for Γ ⊂ PSL2(R) a fuchsian group of the first kind, equipped with the Poincaré metric. This metric is singular at cusps and elliptic fixed points, and the original results of Deligne and Gillet-Soulé do not apply to this setting. Our theorem relates the determinant of cohomology of the trivial sheaf, with an explicit Quillen type metric in terms of the Selberg zeta function of Γ, to a metrized version of the ψ line bundle of the theory of moduli spaces of pointed orbicurves, and the self-intersection bundle of a suitable twist of the canonical sheaf ωX. We make use of surgery techniques through Mayer-Vietoris formulae for determinants of laplacians, in order to reduce to explicit evaluations of such for model hyperbolic cusps and cones. We carry out these computations, that are of independent interest: we provide a rigorous method that fixes incomplete computations in theoretical physics, and that can be adapted to other geometries. We go on to derive an arithmetic Riemann-Roch formula in the realm of Arakelov geometry, that applies in particular to integral models of modular curves with elliptic fixed points. This vastly extends previous work of the first author, whose deformation theoretic methods were limited to the presence only of cusps. As an application, we treat in detail the case of the modular curve X(1), that already reveals the interesting arithmetic content of the metrized ψ line bundles. From this, we solve the longstanding question of evaluating the Selberg zeta special value Z (1, PSL2(Z)). The result is expressed in terms of logarithmic derivatives of Dirichlet L functions. In the analogy between Selberg zeta functions and Dedekind zeta functions of number fields, this formula can be seen as the analytic class number formula for Z(s, PSL2(Z)). The methods developed in this article were conceived so that they afford several variants, such as the determinant of cohomology of a flat unitary vector bundle with finite monodromies at cusps. Our work finds its place in the program initiated by Burgos-Kramer-Kühn of extending arithmetic intersection theory to singular hermitian vector bundles

From natural numbers to quaternions

This textbook offers an invitation to modern algebra through number systems of increasing complexity, beginning with the natural numbers and culminating with Hamilton's quaternions. Along the way, the authors carefully develop the necessary concepts and methods from abstract algebra: monoids, groups, rings, fields, and skew fields. Each chapter ends with an appendix discussing related topics from algebra and number theory, including recent developments reflecting the relevance of the material to current research. The present volume is intended for undergraduate courses in abstract algebra or elementary number theory. The inclusion of exercises with solutions also makes it suitable for self-study and accessible to anyone with an interest in modern algebra and number theory

From Natural Numbers to Quaternions

This textbook offers an invitation to modern algebra through number systems of increasing complexity, beginning with the natural numbers and culminating with Hamilton's quaternions. Along the way, the authors carefully develop the necessary concepts and methods from abstract algebra: monoids, groups, rings, fields, and skew fields. Each chapter ends with an appendix discussing related topics from algebra and number theory, including recent developments reflecting the relevance of the material to current research. The present volume is intended for undergraduate courses in abstract algebra or elementary number theory. The inclusion of exercises with solutions also makes it suitable for self-study and accessible to anyone with an interest in modern algebra and number theory